Second Moments or Moments of Inertia

Transcription

1 Second Moments or Moments of Inertia The second moment of inertia of an element of area such as da in Figure 1 with respect to any axis is defined as the product of the area of the element and the square of the distance from the axis to the element. Rectangular and Polar Moments of Inertia Consider the area A in the x-y plane, Fig. 1. The moments of inertia of the element da about the x- and y-axes are, by definition, di x = y 2 da,and di y = x 2 da respectively. The moments of inertia of A about the same axes are therefore I x = y 2 da Figure 1 I y = x 2 da where we carry out the integration over the entire area The second moment of the element of the figure 1 with respect to an axis through O perpendicular to the plane of area is dj o = r 2 da = (x 2 + y 2 )da The polar moment of inertia of the area is J o = (x 2 + y 2 ) da = x 2 da + y 2 da = I x + I y Thus, the polar moment of inertia of an area is equal to the sum of the rectangular moments of inertia with respect to any two perpendicular axes intersecting the polar axis. The dimensions of moments of inertia of areas are clearly L 4, where L stands for the dimension of length. Thus, the SI units for area moments of inertia are expressed as quartic meters (m 4 ) or quartic millimeters (mm 4 ). The U.S. customary units for area moments of inertia are quartic feet (ft 4 ) or quartic inches (in. 4 ). 1

2 The moment of inertia of an element involves the square of the distance from the inertia axis to the element. Consequently, the area moment of inertia about any axis is always a positive quantity. In contrast, the first moment of the area, which was involved in the computations of centroids, could be either positive, negative, or zero. The Parallel-Axis Theorem for an Area The parallel-axis theorem can be used to find the moment of inertia of an area about any axis that is parallel to an axis passing through the centroid and about which the moment of inertia is known. The parallel-axis theorem (sometimes called the transfer formula). The parallel-axis theorem can he stated as follows: The moment of inertia of an area with respect to any axis is equal to the moment of Inertia with respect to a parallel axis through the centroid of the area plus the product of the area and the square of the distance between the two axes. Thus, I b = Ad 2 + I c Where Ic is the second moment of the area with respect to an axis through the centroid parallel to the axis b, A is the area, and d is the distance between the two axes. Similarly, J b = Ad 2 + J c Second Moments of Areas by Integration In determining the moment of inertia of an area by integration, it is necessary to select an element of an area where all parts of the element selected are the same distance from the moment axis, the moment of inertia of the element of area is obtained directly from the definition of a second moment as the product of the square of this distance and the area of the element. Example: Determine the moments of inertia of the rectangular area about the centroidal x0 and y0-axes, the centroidal polar axis z0 through C, the x- axis, and the polar axis z through O. For the calculation of the moment of inertia about the x0-axis, a horizontal strip of area b dy is chosen so that all elements of the strip have the same y-coordinate. Thus, 2

3 By interchange of symbols, the moment of inertia about the centroidal y0-axis is The centroidal polar moment of inertia is By the parallel-axis theorem the moment of inertia about the x-axis is We also obtain the polar moment of inertia about O by the parallel-axis theorem, which gives us Moments of Inertia of Composite Areas A composite area consists of two or more simple areas, such as rectangles, triangles, and circles. The cross-sectional areas of standard structural elements, such as channels, I beams, and angles, are frequently included in composite areas. The moment of inertia of a composite area with respect to any axis is equal to the sum of the moments of inertia of its component areas with respect to the same axis. When an area, such as a hole, is removed from a larger area, its moment of inertia is subtracted from the moment of inertia of the larger area to obtain the net moment of inertia. Table 1 illustrates Moment of inertia of common geometric areas 3

4 Table 1: Moment of inertia of common geometric areas 4

5 Example: Determine the moments of inertia about the x- and y-axes for the shaded area. The given area is subdivided into the three subareas shown a rectangular (1), a quarter-circular (2), and a triangular (3) area. Two of the subareas are holes with negative areas. 5